Download presentation

Presentation is loading. Please wait.

Published byAllan Sparks Modified over 6 years ago

1
4.3 Logarithm Functions Recall: a ≠ 1 for the exponential function f(x) = a x, it is one-to-one with domain (-∞, ∞) and range (0, ∞). when a > 1, it is always increasing when 0 < a < 1, it is always decreasing In both cases, the exponential function has an inverse which is called the Logarithm Function with base a. The Logarithm Function with Base a For each positive number a ≠ 1 and each x in (0, ∞), and for each x in (0, ∞) and each real number y.

2
Ex 1: Use the inverse relationship with the exponential functions to determine x in the following. Solution: a.) From the definition, the conversion implies that (x – 4)(x + 2), so x = 4 or x = 2

3
Note: The logarithm function is the inverse of the exponential function. It is the graph of the exponential function reflected over the line y = x. Graph on board: Note: The inverses of the exponential functions intersect at (1, 0) and x = 0 is a vertical asymptote. Ex 2: Sketch the graphs of the following. is the inverse of the function is still the inverse of the function We shift one unit right then 3 units up.

4
Arithmetic Properties of Logarithms: For each positive number a ≠ 1, each pair of positive real numbers x 1, x 2 and each real number r we have, Note: Just as exponential functions have inverses, so does the natural exponential function. (e).

5
The Natural Logarithm Function For each x in (0, ∞), we have, for each x in (0, ∞) and each real number y. Arithmetic Properties of Natural Logarithms For each positive number a ≠ 1, each pair of positive real numbers x 1, x 2, and each real number r we have,

6
Ex 3: Determine the values of x that satisfy each expresssion. Solution: a.) The product rule implies that… Taking the inverse of both sides (e) cancels out the “ln” leaving… Simplify…1 = x – 2 so, x = 3

7
b.) Here, we use the product and exponent rules ln 3 + ln (2x – 1) = ln 3(2x – 1) = ln (6x – 3) and… So we now have… Taking the inverse of both sides…

8
c.) We can rewrite the equation as… Using the arithmetic properties… Changing this to its equivalent exponential form… So… x = 5 or x = -1

9
d.) We can rewrite this equation as… Multiply both sides by 2 Take inverse of both sides Solve for x You could verify that this is a solution by substituting in back into the original; you will see it equals ½.

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google